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Shown is a regular octagon, which region has a greater area?

 

 Dec 21, 2019
 #1
avatar+111360 
+2

Let the side of the octagon  = S

 

And  one of the yellow  regions  is a trapezoid

And the  height of one of these trapezoids =  S / √2

And the base  of one of these trapezoids =   S  + 2S/√2 =   (1 +√2)S

 

So....the area of one of the trapezoids =  (1/2) (S/√2) ( S + (1 + √2) S)  =

(√2/4) ( 2 + √2)S^2  =

(1/4) (2 + 2√2)S^2  =

(1/2) (  1 + √2)S^2

So  the two trapezoids  have an  area  of  ( 1 + √2)S^2 =  the yellow area

 

 

And the area of the blue rectangle =   S  * Base of the trapezoid =

S *  (1 + √2)S  =  (1 + √2)S^2

 

So.....the  yellow and blue areas are equal  !!!!

 

cool cool cool

 Dec 21, 2019
 #2
avatar+109497 
+1

I never would have guessed that, thanks Chris.

Melody  Dec 21, 2019

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